Copied to
clipboard

G = C2xC4.10C42  order 128 = 27

Direct product of C2 and C4.10C42

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2xC4.10C42, C24.6Q8, (C2xC4).88C42, C4.41(C2xC42), (C22xC8).17C4, C23.49(C2xQ8), (C22xC4).34Q8, C23.58(C4:C4), (C22xC4).253D4, C4o(C4.10C42), (C2xM4(2)).24C4, (C23xC4).216C22, (C22xC4).645C23, (C22xM4(2)).9C2, C4.14(C2.C42), (C2xM4(2)).294C22, C22.27(C2.C42), (C2xC8).8(C2xC4), C22.7(C2xC4:C4), (C2xC4).225(C2xD4), C4.79(C2xC22:C4), (C2xC4).120(C4:C4), (C2xC4).515(C22xC4), (C22xC4).478(C2xC4), C2.8(C2xC2.C42), (C2xC4).112(C22:C4), SmallGroup(128,463)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C2xC4.10C42
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C23xC4 — C22xM4(2) — C2xC4.10C42
Lower central
C1 — C4 — C2xC4.10C42
Upper central
C1 — C2xC4 — C2xC4.10C42
Jennings
C1 — C2 — C2 — C22xC4 — C2xC4.10C42

Generators and relations for C2xC4.10C42
 G = < a,b,c,d | a2=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 276 in 186 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C23, C23, C23, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C24, C22xC8, C2xM4(2), C2xM4(2), C23xC4, C4.10C42, C22xM4(2), C2xC4.10C42
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C23, C42, C22:C4, C4:C4, C22xC4, C2xD4, C2xQ8, C2.C42, C2xC42, C2xC22:C4, C2xC4:C4, C4.10C42, C2xC2.C42, C2xC4.10C42

Smallest permutation representation of C2xC4.10C42
►On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)

(1 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)

(1 29 7 31 5 25 3 27)(2 24 4 22 6 20 8 18)(9 17 15 19 13 21 11 23)(10 28 12 26 14 32 16 30)
 
G:=sub<Sym(32)| (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,29,7,31,5,25,3,27)(2,24,4,22,6,20,8,18)(9,17,15,19,13,21,11,23)(10,28,12,26,14,32,16,30)>;
 
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,29,7,31,5,25,3,27)(2,24,4,22,6,20,8,18)(9,17,15,19,13,21,11,23)(10,28,12,26,14,32,16,30) );
 
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28)], [(1,15,5,11),(2,16,6,12),(3,9,7,13),(4,10,8,14),(17,31,21,27),(18,32,22,28),(19,25,23,29),(20,26,24,30)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,29,7,31,5,25,3,27),(2,24,4,22,6,20,8,18),(9,17,15,19,13,21,11,23),(10,28,12,26,14,32,16,30)]])
 

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J8A···8X
order12222···244444···48···8
size11112···211112···24···4

44 irreducible representations

dim111112224
type++++--
imageC1C2C2C4C4D4Q8Q8C4.10C42
kernelC2xC4.10C42C4.10C42C22xM4(2)C22xC8C2xM4(2)C22xC4C22xC4C24C2
# reps14312126114

Matrix representation of C2xC4.10C42 ►in GL6(F17)

1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
0160000
0013000
0001300
0000130
0000013
,
10160000
1670000
000010
000001
0013000
000400
,
0130000
400000
000100
004000
0000013
000010

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[10,16,0,0,0,0,16,7,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,13,0] >;
 

C2xC4.10C42 in GAP, Magma, Sage, TeX 

C_2\times C_4._{10}C_4^2
 
% in TeX
 
G:=Group("C2xC4.10C4^2");
 
// GroupNames label
 
G:=SmallGroup(128,463);
 
// by ID
 
G=gap.SmallGroup(128,463);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,248,1411,172,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=1,c^4=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
 
// generators/relations
 

׿
x
:
Z
F
o
wr
Q
<